High School

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------------------------------------------------ Add the polynomials [tex]$7x^3 + 2x^2 - 5x - 7$[/tex] and [tex]$-5x^2 + x^3 + 4x - 5$[/tex].

Answer :

Sure! Let's add the two polynomials step by step.

We are given two polynomials:
1. [tex]\( 7x^3 + 2x^2 - 5x - 7 \)[/tex]
2. [tex]\( x^3 - 5x^2 + 4x - 5 \)[/tex]

To add these polynomials, we will combine like terms. Like terms are terms that have the same variable raised to the same power.

1. Combine the [tex]\( x^3 \)[/tex] terms:
- [tex]\( 7x^3 \)[/tex] from the first polynomial
- [tex]\( x^3 \)[/tex] from the second polynomial

Adding these, [tex]\( 7x^3 + x^3 = 8x^3 \)[/tex]

2. Combine the [tex]\( x^2 \)[/tex] terms:
- [tex]\( 2x^2 \)[/tex] from the first polynomial
- [tex]\(-5x^2\)[/tex] from the second polynomial

Adding these together, [tex]\( 2x^2 - 5x^2 = -3x^2 \)[/tex]

3. Combine the [tex]\( x \)[/tex] terms:
- [tex]\(-5x\)[/tex] from the first polynomial
- [tex]\(4x\)[/tex] from the second polynomial

Adding these, [tex]\(-5x + 4x = -x\)[/tex]

4. Combine the constant terms:
- [tex]\(-7\)[/tex] from the first polynomial
- [tex]\(-5\)[/tex] from the second polynomial

Adding these, [tex]\(-7 - 5 = -12\)[/tex]

Now, put all the terms together:

The sum of the two polynomials is [tex]\[ 8x^3 - 3x^2 - x - 12 \][/tex]

That's the final simplified result!